1995 American Control Conference. June 21-23, Seattle, Washington. Using Neural Networks. Stanford University.

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1 99 American Control Conference. June -, 99. Seattle, Washington Recongurable Control of a Free-Flying Space Using Neural s Edward Wilson Stephen M. Rock Stanford University Aerospace ics Laboratory Stanford, California 9 ed,rock@sun-valley.stanford.edu Abstract An experimental demonstration of a new, recongurable neural-network-based adaptive control system is presented. The system under control is a laboratory model of a free-ying space robot whose position and attitude are controlled using eight on-o air thrusters. The neural-network controller adapts in real-time to account for multiple destabilizing thruster failures. The adaptive control system is similar in structure to a conventional indirect adaptive control system. While a system identication process builds a model of the robot, a neural controller is trained concurrently using backpropagation to optimize performance using this model. The active controller is updated every few seconds, yielding quick adaptation. Stability is restored within seconds, system identication is complete within seconds, and near-optimal performance is achieved within minutes.. Introduction The nonlinear, parallel, and adaptive capabilities of neural networks make them promising for control applications. Neural networks derive their advantage in solving very complex problems from the emergent properties that come with the massive interconnection of simple processing units. With good training techniques, the networks are capable of implementing very complex behaviors. Numerous examples in the literature demonstrate the potential of neural-network control []. There are three important issues which often arise in a realworld control application that have not been eectively addressed in the NN literature, however:. A priori knowledge is often available in the form of models of the system's key components and a preliminary control design (e.g., provided Ph.D. Candidate, Department of Mechanical Engineering. Research supported by NASA and AFOSR. Associate Professor, Department of Aeronautics and Astronautics. by \conventional" control design techniques). Is it possible to use this a priori information to improve greatly the performance the neural network can then enable?. Many control applications involve the use of discrete-valued devices. For example, thrusters often operate \on-o" rather than with analogvalued outputs. This presents a problem for backpropagation learning, since these discretevalued functions are not continuously dierentiable. Is it possible to modify backpropagation to accommodate the discrete-valued functions?. Speed of learning is often important in realtime control applications. Can backpropagation learning be made fast enough to be feasible for rapid on-line adaptation? In addition to the development of a recongurable thruster control system, the goal of the work reported here was to develop extensions to neural-network theory that would address each of these issues. These developments are reported in the context of the robot control application that follows.. Control Application The control task addressed in this research is the control of position and attitude of a free-ying space robot using on-o thrusters. Control using on-o thrusters is an important problem for real spacecraft [], and the nonlinear and adaptive capabilities of neural networks make them attractive for this application. A NN-based approximation method scales well to higher-dimensional thruster controllers, as well as providing a structure conducive to recon- gurable control. Further details regarding the robot control application are presented in [] []. The challenge presented here is to damage mechanically a number of thrusters (as in Figure ), and then have the control system autonomously and rapidly recongure itself in real time. Some thruster failures are strongly destabilizing, which places high demands on the speed of recovery.

2 The three degrees of freedom (x; y; ) of the base are controlled using eight thrusters positioned around its perimeter, as shown in Figure. Each thruster produces both a torque and net force on the robot. This coupling, combined with the on-o nature of the thrusters, substantially complicates the control task. The robot-base-control strategy developed for this system is shown in Figure. The thruster mapping task that must be performed during each sample period is to take an input vector of continuous-valued desired forces and torques, [F xdes;f ydes; des], and nd the optimal output vector of discrete-valued (o, on) thruster values, [T ;T ; :::; T ]. desired state vector, Xdes X PD controller desired force vector, Fdes.9 N -. N [. N-m] Thruster Mapper (NN) thruster pattern, T [ ] Position Sensor Figure : Stanford Free-Flying Space This highly-autonomous mobile robot operates in the horizontal plane, using an air-cushion suspension to simulate the drag-free and zero-g characteristics of space. The experimental equipment, shown in Figure, is a fully-self-contained planar laboratory-prototype of an autonomous free-ying space robot complete with on-board gas, thrusters, electrical power, multiprocessor computer system, camera, wireless Ethernet data/communications link, and two cooperating manipulators. It exhibits nearly frictionless motion as it oats above a granite surface plate on a micron thick cushion of air []. Accelerometers and an angular-rate sensor sense base motions. Nominal Configuration 6 7 After Multiple Failures 6 7 Figure : Example failure mode Magnitude and direction of each of the eight thrusters is shown. Thruster failures were simulated mechanically with weaker thrusters and and 9 elbows. Some elbows destabilize the robot by changing the sign of the thrust in the direction. Figure : -Base Control The PD control module treats the thrusters as continuous actuators. The thruster mapper must nd the thruster pattern that will produce a force closest to that requested by the base-control module. In [], a conventional approach is presented that solves this problem by performing an exhaustive search atevery sample period. With eight bi-level thrusters, 6 possible thruster combinations exist. By clever use of symmetries, this search space was reduced to, making the problem tractable. Unfortunately, this solution method does not scale well for a three-dimensional robot, or when thruster failures are allowed, disrupting the symmetries. This provides the motivation for using a neural network: the neural network is used to implement a nonlinear approximation to the optimal solution { one that can be computed in real-time.. Recongurable Control System Often the most important, and sometimes the most dicult aspects of a neural-network control application are the decisions about how to structure the control system and which components are to be neural-network-based. The rst issue is to determine whether the application is one where neural networks can contribute eciently better (and cheaper) control than is achievable without them. If they can, the second issue is to determine in just which segment(s) of the control system they should be used in order to do so.

3 To determine where neural networks can contribute eectively, the control systems engineer must consider the strengths of neural networks (nonlinear, adaptive, generic, unstructured, parallelizable) as well as the costs associated with these benets (dicult to understand workings or prove stability, design is iterative, computationally complex). The cost/benet balance must be evaluated on an application by application basis. First at the system level, the system requirements and considerations of degree-ofnonlinearity, adaptation requirements, and computational complexity, etc., lead to a candidate system architecture. Then at the component level, this cost/benet analysis is repeated, leading to the decision of what sort of subsystem will be used in each segment of the control system []. User Trajectory Generator Xdes X PD controller NN train Fdes Thruster T Mapper (NN).9 N -. N [. N-m] [ ] Adaptive System robot model ID Accel. accel Sensors x,y,ψ Position Sensor Figure : Recongurable Control System This control system is based upon a conventional indirect adaptive controller, such as a self-tuning regulator. The ID block represents a recursive least squares identication of thruster strength and direction. This continually-updated model is passed to the NN training block, shown in detail in Figure. The continually-updated neural thruster mapper is loaded into the active control loop every few seconds. Applying these principles to the robot control application has resulted in a structure, shown in Figure, that is modelled after a standard control architecture known as \indirect adaptive control." \Indirect" refers to the use of sensor information to build a model of the system, and then to redesign a controller based upon the updated plant model. A recursive linear regression component was used for failure detection and identication, since identication of the thruster characteristics is a linear process. The algorithm used to obtain acceleration measurements was nonlinear, but could be derived analytically, so no neural network was used there either. A neural network was used precisely at the location where it is benecial: the thruster mapper. This is an inscrutable nonlinear function that requires adaptation, and the benets of a neural-network approach do indeed outweigh the costs. The control redesign process is therefore a backpropagation-based neuralnetwork training algorithm.. Neural Developments In development of the control system, two major neural network challenges were faced that resulted in development of a new NN architecture and training method. These developments are mentioned briey here, and in more detail in [] [] [6]... Fully-Connected Architecture To speed up the reconguration process, a general neural-network architecture was developed. This \Fully-Connected Architecture" is for feedforward neural networks that can be trained using backpropagation [7]. The FCA has many advantages over a layered architecture [] [6]. For this application, the most signicant advantage comes from the feedthrough weights. These weights provide a direct, linear connection matrix from inputs to outputs (provided sigmoids are used only on hidden units). This produces fast initial learning, and allows direct pre-programming of a linear solution calculated by some other method. This is especially important for control applications, where a large body of linear control knowledge exists that can be drawn upon to provide a good starting point. The FCA provides a seamless integration of linear and nonlinear components. In this application, a linear approximate solution is calculated very quickly based on a pseudo-inverse of a linearized plant model. Injecting this approximate linear solution into the network immediately after a failure is detected results in rapid stabilization of the robot... Backpropagation Learning for Discrete-Valued Functions Training a neural network to produce a thruster mapping based upon a model of the robot can be thought of as learning the inverse model of the robot-thruster system, as in Figure. This is a common approach, and would be relatively straightforward if not for the discrete-valued functions that represent the ono thrusters. Some modication to the learning algorithm is required to allow gradient-based optimization to be used with these non-dierentiable functions. The method, shown in Figure, relies on approximation of the discrete-valued functions with \noisy sigmoids" during training. This is a broadly-applicable

4 RUN-TIME Fdesired Neural Tcontinuous Tdiscrete Factual Model TRAINING Forward Sweep Fdes Td Fact Neural Tc Σ Σ Model N Backward Sweep Neural Tc N Td Fact Td ε Fact ε Fdes ( ) ε cost = Figure : Thruster mapping, on-line training method This is the training structure used to adapt the thruster mapper during reconguration. This process appears as the \NN train" block in Figure. The \ Model" contains the magnitude and direction of each thruster. During adaptation, this model is updated continually by the \ID" process shown in Figure. algorithm that applies to any gradient-based optimization involving discrete-valued functions. It is described in detail in [] [].. Experimental Demonstration Position and attitude of the robot base are controlled while subject to multiple, large, possibly-destabilizing changes in thruster characteristics. An o-board vision system provides high-bandwidth position feedback, which is then digitally ltered and dierentiated to provide velocity feedback. On-board accelerometers and an angular-rate sensor are used to provide base-acceleration measurements used by the failuredetection and control-reconguration capability. The FCA neural-network thruster-mapping component described in Section was implemented on the on-board Motorola 6 processor, as was the rest of the control system (at a Hz sample rate). Figure shows the robot thruster layout in nominal and failed congurations. The magnitude and direction of each thruster is shown. Nominally, each thruster produces Newton of force, directed as shown. The failures were produced by mechanically changing the thrusters. Failures include: halfstrength, plugged completely, angled at, and angled at 9. The 9 failure mode places high demands on the control reconguration system, since it destabilizes the robot (changing the direction of torque results in positive feedback)... Failure Detection Accelerometers and an angular-rate sensor (that is digitally dierentiated) produce acceleration signals in (x; y; ). These signals are ltered and passed to a system identication process based upon recursive least squares. The parameters identied are the accelerations in (x; y; ) resulting from each thruster ring. When a failure is detected, the thruster is excited articially to speed up the identication process. For the case presented here, with 6 of thrusters failed, the ID process took about seconds from when the rst thruster red until the last thruster was identied to a high level of condence. Since the neural network is training in parallel with this process, stabilization occurs within seconds, and the robot remains well-controlled during the identication. Results from the reconguration are shown in Figure 6. The static control deadband is approximately cm in translation and in rotation. Due to noisy sensors and multiple thrusters ring simultaneously, it takes about seconds before the ID process is condent enough to conrm a thruster failure and begin reconguration. An initial stabilizing reconguration occurs nearly instantaneously as a linear approximate solution is quickly calculated and implemented via the Fully Connected Architecture. Once failures have been detected, they are labelled as \suspects" and are excited articially to expedite identication. The neural-network training process shown in Figure runs in parallel with the ID process, with the robot model updated continually by the ID. The network being trained is copied periodically into the network that is controlling the robot... Rapid Reconguration Very rapid learning is possible due rst to the FCA, and due second to the growing of the network. With few hidden neurons, quick learning takes place since fewer computations are required, and fewer training patterns are required (to avoid overtting). The network begins with inputs, hidden neurons, and outputs, and gradually grows to or more hidden neurons as training progresses. New hidden neurons are added when performance begins to plateau. To prevent overtting, the training-set size is grown proportionally with the number of hidden neurons. With this arrangement, a mapping with about % error above optimal results in seconds, % above optimal within 6 seconds, and % above optimal Due to the use of discrete-valued actuators, there is almost always a force error vector. The error value reported here indicates that the average magnitude of the force error vector is. times the magnitude achievable with an exhaustive search.

5 error [cm,deg] Thruster Number 7 6 neural-network training under way training with knowledge of all failures artificial excitation, ID 6 7 time [sec] Six thrusters are severely misconfigured, as in figure t <, robot within deadband, no thrusters firing t =, small disturbance applied to robot t =, thrusters,,6 suspected, stabilizing mapper loaded, training begins t = 6, sixth and final thruster failure confirmed to high level of confidence t =, all thruster characteristics confirmed t, neural-network optimization continues 6 7 time [sec] thruster is a candidate for artificial excitation thruster failure is suspected, but not yet confirmed X axis Y axis Yaw angle Figure 6: Reconguration Experiment, Position Errors, Thruster Firing [x; y; ] position errors and thruster signals are plotted during autonomous reconguration. Black rectangular regions indicate periods of thruster ring. Darkly-shaded regions indicate the time during which the thruster was suspected. In addition to articial excitation of the suspected thrusters, excitation of other thrusters is used to expedite the identication process. These periods are indicated by the lightly-shaded regions. The robot begins at rest within the deadband, is disturbed at t =, stabilizes itself within seconds, and completes identication (aided with articial excitation) after seconds. The neuralnetwork thruster mapper continues to optimize after the identication is complete. within seconds. As more hidden neurons are added, the network performance approaches optimality, but at the expense of a slower training rate. 6. Summary and Conclusions This paper has presented an adaptive neural-networkbased control system for a free-ying space robot. This control system has a structure that is modelled after a conventional indirect adaptive controller, with the neural network used to implement the nonlinear adaptive component. Specic procedures for determining neural-network applicability are outlined in the paper. Two neural-network-control developments were critical to achieving quick adaptation and a near-optimal controller. First, a \Fully-Connected Architecture" (FCA) was used that has the ability to incorporate an a priori approximate linear solution instantly; this permits quick stabilization by an approximate linear controller. Second, a learning method was used that allows gradient-based optimization (backpropagation) with discrete-valued functions, in this case, the on-o thrusters. The control system was demonstrated experimentally on a laboratory prototype robot, where stable recon- guration to major destabilizing thruster failures occurred within seconds, and near-optimal control was achieved within minutes. References [] W. Thomas Miller III, Richard S. Sutton, and Paul J. Werbos, editors. Neural s for Control. Neural Modeling and Connectionism. The MIT Press, Cambridge, MA, 99. [] James R. Wertz, editor. Spacecraft Attitude Determination and Control. Kluwer Academic Publishers, Boston, 97. [] Edward Wilson and Stephen M. Rock. Neural network control of a free-ying space robot. Simulation, June 99. [] Edward Wilson. Experiments in Neural Control of a Free-Flying Space. PhD thesis, Stanford University, Stanford, CA 9, March 99. [] Marc A. Ullman. Experiments in Autonomous Navigation and Control of Multi-Manipulator, Free- Flying Space s. PhD thesis, Stanford University, Stanford, CA 9, March 99. [6] Edward Wilson and Stephen M. Rock. Experiments in control of a free-ying space robot using fully-connected neural networks. In Proceedings of the World Congress on Neural s, volume, pages 7{6, Portland OR, July 99. [7] Paul J. Werbos. Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. PhD thesis, Harvard University, Cambridge, MA, August 97. [] Edward Wilson. Backpropagation learning for systems with discrete-valued functions. In Proceedings of the World Congress on Neural s, volume, pages {9, San Diego CA, June 99. International Neural Society.

Fimp xcmd xx Imp. Law Des EOM. Flex Kinematics. F ext. Flex EOM. Flex EOM. f 2. τ 2. Arm 2 CT Controller. f 1. τ 1. f cmd. Arm 1 CT Controller.

Fimp xcmd xx Imp. Law Des EOM. Flex Kinematics. F ext. Flex EOM. Flex EOM. f 2. τ 2. Arm 2 CT Controller. f 1. τ 1. f cmd. Arm 1 CT Controller. Proceedings of the International Conference on Robotics and Automation, IEEE, May 994 Experiments in Object Impedance Control for Flexible Objects David W. Meer Stephen M. Rock y Aerospace Robotics Laboratory